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Re: power of logistic distribution
Posted:
Jun 18, 2012 5:41 AM
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dist = LogisticDistribution[m, s];
Simplify[Moment[dist, n], n >= 0]
(2*Pi)^n*((-I)*s)^n*BernoulliB[n, 1/2 + (I*m)/(2*Pi*s)]
And@@Table[Moment[dist, n] == Mean[TransformedDistribution[z^n, z\[Distributed]dist]], {n, 0, 40}]//Simplify
True
Bob Hanlon
On Jun 17, 2012, at 3:58 AM, paul <paulvonhippel@yahoo.com> wrote:
> I would like an expression for the mean of a variable that is some integer power of a logistic variable. I have tried the following approach, which did not work. Many thanks for any suggestions.
>
> Here is what I've done so far. If I specify the power (e.g., power=2), Mathematica returns an answer rather quickly -- e.g.,
>
> In[66]:= Mean[TransformedDistribution[Z^2,
> Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
> Out[66]= 1/3 (3 \[Mu]^2 + \[Pi]^2 \[Sigma]^2)
>
> I get quick results if the power is 2, 3, 4, ..., 100. So it seems to me there must be some general solution for integer powers. But when I ask for that general solution, Mathematica simply echoes the input:
>
> In[64]:= Assuming[p \[Element] Integers,
> Mean[TransformedDistribution[Z^p,
> Z \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]]
> Out[64]= Mean[
> TransformedDistribution[\[FormalX]^
> p, \[FormalX] \[Distributed] LogisticDistribution[\[Mu], \[Sigma]]]]
>
> Why isn't it giving me something more digested?
>
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